infer
Class: regARIMA
Infer innovations of regression models with ARIMA errors
Syntax
E = infer(Mdl,Y)
[E,U,V,logL]
= infer(Mdl,Y)
[E,U,V,logL]
= infer(Mdl,Y,Name,Value)
Description
E = infer(Mdl,Y) infers
residuals of a univariate regression model with ARIMA time series
errors fit to response data Y.
[E,U,V,logL]
= infer(Mdl,Y) additionally
returns the unconditional disturbances U, the innovation
variances V, and the loglikelihood objective function
values logL.
[E,U,V,logL]
= infer(Mdl,Y,Name,Value) returns
the output arguments using additional options specified by one or
more Name,Value pair arguments.
Mdl 
Regression model with ARIMA errors, as created by regARIMA or estimate.
The parameters of Mdl cannot contain NaNs.

Y 
Response data, specified as a column vector or numObsbynumPaths matrix. infer infers
the innovations and unconditional disturbances of Y. Y represents
the time series characterized by Mdl, and is the
continuation of the presample series Y0. If Y is a column vector, then it
represents one path of the underlying series. If Y is a matrix, then it represents numObs observations
of numPaths paths of an underlying time series.
infer assumes that observations across any
row occur simultaneously. The last observation of any series is the
latest.

NameValue Pair Arguments
Specify optional commaseparated pairs of Name,Value arguments.
Name is the argument
name and Value is the corresponding
value. Name must appear
inside single quotes (' ').
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN.
'E0' 
Presample innovations that have mean 0 and provide initial values
for the ARIMA error model, specified as the commaseparated pair consisting
of 'E0' and a column vector or matrix. If E0 is a column vector, then
it is applied to each inferred path. If E0 is a matrix, then it requires
at least numPaths columns. If E0 contains
more columns than required, then infer uses the
first numPaths columns. E0 must contain at least Mdl.Q rows.
If E0 contains more rows than required, then infer
uses the latest presample innovations. The last row contains the latest
presample innovation.
Default: infer sets the necessary observations to
0. 
'U0' 
Presample unconditional disturbances that provide initial values
for the ARIMA error model, specified as the commaseparated pair consisting
of 'U0' and a column vector or matrix. If U0 is a column vector, then
it is applied to each inferred path. If U0 is a matrix, then it requires
at least numPaths columns. If U0 contains
more columns than required, then infer uses the
first numPaths columns. U0 must contain at least Mdl.P rows.
If U0 contains more rows than required, then infer
uses the latest presample unconditional disturbances. The last row
contains the latest presample unconditional disturbance.
Default: infer backcasts for the necessary presample
unconditional disturbances. 
'X' 
Predictor data in the regression model, specified as the commaseparated
pair consisting of 'X' and a matrix.
The columns of X are separate, synchronized
time series, with the last row containing the latest observations.
The number of rows of X should be at least the
length of Y. If the number of rows of X exceeds
the number required, then infer uses the latest observations.
Default: infer does not include a regression component
in the model regardless of the presence of regression coefficients
in Mdl. 
Notes
NaNs in Y, X, E0,
and U0 indicate missing values and infer removes
them. The software merges the presample data sets (E0 and U0),
then uses listwise deletion to remove any NaNs. infer similarly
removes NaNs from the effective sample data (X and Y).
Removing NaNs in the data reduces the sample size,
and can also create irregular time series. infer assumes that you synchronize
presample data such that the latest observation of each presample
series occurs simultaneously. All predictors (i.e., columns in X)
are associated with each response path in Y. V is equal to the variance in Mdl.

Output Arguments
E 
Inferred residuals (estimated innovations of the unconditional
disturbances), returned as a numObsbynumPaths matrix.

U 
Inferred unconditional disturbances, returned as a numObsbynumPaths matrix.

V 
Inferred variances, returned as a numObsbynumPaths matrix.

logL 
Loglikelihood objective function values associated with the
model specification, returned as a numPathselement
vector. Each element of logL is associated with
the corresponding path of Y.

Examples
expand all
Forecast responses from the following regression
model with ARMA(2,1) errors over a 30period horizon:
where ε_{t} is
Gaussian with variance 0.1.
Specify the regression model with ARIMA errors. Simulate
responses from the model and two predictor series.
Mdl = regARIMA('Intercept', 0, 'AR', {0.5 0.8}, ...
'MA',0.5,'Beta',[0.1 0.2], 'Variance',0.1);
rng(1);
X = randn(100,2);
y = simulate(Mdl,100,'X',X);
Infer, and then plot residuals. By default, infer backcasts
for the necessary presample unconditional disturbances.
E = infer(Mdl,y,'X',X);
figure
plot(E)
title('Inferred Residuals')
Regress the log GDP onto the CPI using a regression
model with ARMA(1,1) errors, and then examine the residuals.
Load the U.S. Macroeconomic data set and preprocess the
data.
load Data_USEconModel;
logGDP = log(Dataset.GDP);
dlogGDP = diff(logGDP); % For stationarity
dCPI = diff(Dataset.CPIAUCSL); % For stationarity
T = length(dlogGDP); % Effective sample size
Fit a regression model with ARMA(1,1) errors.
ToEstMdl = regARIMA(1,0,1);
EstMdl = estimate(ToEstMdl,dlogGDP,'X',dCPI);
Infer the residuals over all observations. By default, infer backcasts
for the necessary unconditional disturbances.
E = infer(EstMdl,dlogGDP,'X',dCPI);
Plot the inferred residuals.
figure
plot(E)
title('{\bf Inferred Residuals}')
The residuals are centered around 0, but show signs of heteroscedasticity.
References
[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time
Series Analysis: Forecasting and Control. 3rd ed. Englewood
Cliffs, NJ: Prentice Hall, 1994.
[2] Davidson, R., and J. G. MacKinnon. Econometric
Theory and Methods. Oxford, UK: Oxford University Press,
2004.
[3] Enders, W. Applied Econometric Time Series.
Hoboken, NJ: John Wiley & Sons, Inc., 1995.
[4] Hamilton, J. D. Time Series Analysis.
Princeton, NJ: Princeton University Press, 1994.
[5] Pankratz, A. Forecasting with Dynamic Regression
Models. John Wiley & Sons, Inc., 1991.
[6] Tsay, R. S. Analysis of Financial Time Series.
2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.
See Also
estimate  forecast  regARIMA  simulate
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